A central potential is a potential that depends only on the absolute value of the distance away from the potential's center. A central potential is rotationally invariant. We may use these properties to reduce this otherwise three-dimensional problem to an effective one-dimensional problem. The general form of the Hamiltonian for a particle immersed in such a potential is
Due to rotational symmetry,
and
This allows us to find a complete set of states that are simultaneous eigenstates of
and
We will label these eigenstates as
where
and
are the orbital and magnetic quantum numbers, as defined in the previous chapter, and
represents the quantum numbers that define the radial dependence of the wave function; this is the only part of the state that depends on the exact form of the potential, as we will see shortly.
Let us now write the Schrödinger equation for this system and solve for the angular dependence of the wave function, thus reducing the problem to an effective one-dimensional problem. The equation is
The Laplacian in spherical coordinates may be written as
so that the equation becomes
We already know the eigenfunctions of
from the previous chapter, and thus the entire angular dependence of the wave function. We may therefore use separation of variables and write
where
are the spherical harmonics. Substituting this into the Schrödinger equation, we obtain
The term
is referred to as the centrifugal barrier, which is associated with the motion of the particle. The classical analogue is the term,
that arises in treating central forces classically. The centrifugal barrier prevents the particle from reaching the center of force, causing the wave function to vanish at this point.
Now, if we let
, we finally arrive at the effective one-dimensional Schrödinger equation for the radial dependence of the wave function,
Due to the boundary condition that
must be finite the origin,
must vanish.
In many cases, looking at the asymptotic behavior of
can be quite helpful, as we will see in later sections.
Nomenclature
Historically, the first four (previously five) values of
have taken on names, and additional values of
are referred to alphabetically:
